Here's how to find the equation for the distance an object falls before reaching terminal velocity and estimate the distance for a piece of notebook paper

1. Combining Equations:

We are given two equations

Speed due to gravity: |v| = gt (where v is the object's speed, g is acceleration due to gravity, and t is time)

Terminal velocity: v_terminal = √(2mg / ρA) (where m is the object's mass, ρ is air density, A is the object's cross-sectional area, and C_D is the drag coefficient, assumed to be ≈ 1)

We want to find the time (t) it takes for the object's speed to reach terminal velocity (v_terminal).

2. Solving for Time:

Since the object accelerates constantly until reaching terminal velocity, we can set its speed (v) equal to the terminal velocity (v_terminal) in the first equation:

v_terminal = gt

Substitute the expression for terminal velocity from the second equation:

√(2mg / ρA) = gt

3. Square Both Sides (be cautious):

Square both sides to get rid of the square root (remembering that squaring introduces extraneous solutions, so we'll need to check for those later):

2mg / ρA = g^2 * t^2

4. Isolate Time:

Solve for time (t):

t = √(2mg / (ρA * g^2))

5. Estimating Distance:

Once we have the time (t), we can estimate the distance (d) the object falls by multiplying the terminal velocity (v_terminal) by the time (t):

d = v_terminal * t = √(2mg / ρA) * √(2mg / (ρA * g^2))

Simplify the equation:

d = √((2mg)^2 / (ρA * g^2 * ρA))

Cancel out common factors and remove the square root (since distance cannot be negative):

d = 2m / (ρA)

6. Estimating for Notebook Paper (as an example):

Let's estimate the distance for a piece of notebook paper (assuming no wind resistance and the chosen estimates for mass, density, and area):

Mass of a sheet of notebook paper (m): Assume m ≈ 0.005 kg (This is an estimate, the actual mass can vary)

Air density (ρ): ρ ≈ 1.2 kg/m³ (at sea level)

Area of a sheet of notebook paper (A): Assume a typical notebook paper size of 21.6 cm x 27.9 cm. Convert to meters: A ≈ 0.06 m x 0.08 m = 0.0048 m²

Plug these values into the equation:

d = 2 * 0.005 kg / (1.2 kg/m³ * 0.0048 m²)

d ≈ 0.83 meters